## GENUS DISTRIBUTIONS OF STAR-LADDERS

### BibTeX

@MISC{Chen_genusdistributions,

author = {Yichao Chen and Jonathan L. Gross and Toufik Mansour},

title = {GENUS DISTRIBUTIONS OF STAR-LADDERS},

year = {}

}

### OpenURL

### Abstract

Abstract. Star-ladder graphs were introduced by Gross in his development of a quadratic-time algorithm for the genus distribution of a cubic outerplanar graph. This paper derives a formula for the genus distribution of star-ladder graphs, using Mohar’s overlap matrix and Chebyshev polynomials. Newly developed methods have led to a number of recent papers that derive genus distributions and total embedding distributions for various families of graphs. Our focus here is on a family of graphs called star-ladders. 1.

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Citation Context ...own in Figure 2. Figure 2. The star-ladder SL2,1,0. 1.2. Genus polynomial. It is assumed that the reader is somewhat familiar with the basics of topological graph theory, as found in Gross and Tucker =-=[7]-=-. All graphs considered in this paper are connected. A graph G = (V (G), E(G)) is permitted to have both loops and multiple edges. A surface is a compact 2-manifold without boundary. In topology, surf... |

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Citation Context ...en investigated in the past quarter century, since the topic was inaugurated by Gross and Furst [6]. The contributions include [1, 5, 8, 9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 22, 23, 24, 25, 26] and =-=[27]-=-. Gross [11] presents a quadratic-time algorithm for computing the genus distribution of any cubic outerplanar graph. He analyzes the structure of any cubic outerplanar graph and finds that such a gra... |

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Citation Context ...sed formula for the genus distribution of the cubic outerplanar graphs. Our closed formula in this paper for the genus distribution of star-ladders is derived with the aid of Mohar’s overlap matrices =-=[17]-=-. 1.1. Star-ladders. An n-rung closed-end ladder Ln can be obtained by taking the graphical cartesian product of an n-vertex path with the complete graph K2, and then doubling both its end edges. The ... |

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4 |
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(Show Context)
Citation Context ...re gi(G) means the number of embeddings of G into the orientable surface Si, for i ≥ 0. 1.3. Overlap matrices. Mohar [17] introduced an invariant that has subsequently been used numerous times (e.g., =-=[2, 3, 4]-=-) in the calculation of distributions of graph embeddings, including non-orientable embeddings. We use Mohar’s invariant here in our derivation of a formula for the genus distribution of star-ladders.... |

4 |
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(Show Context)
Citation Context ...re gi(G) means the number of embeddings of G into the orientable surface Si, for i ≥ 0. 1.3. Overlap matrices. Mohar [17] introduced an invariant that has subsequently been used numerous times (e.g., =-=[2, 3, 4]-=-) in the calculation of distributions of graph embeddings, including non-orientable embeddings. We use Mohar’s invariant here in our derivation of a formula for the genus distribution of star-ladders.... |

3 |
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3 |
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Citation Context ...er otherwise. The rank is independent of the choice of a spanning tree. For drawing a planar representation of a rotation system on a cubic graph, we adopt the graphic convention introduced by Gustin =-=[13]-=-, and used extensively by Ringel and Youngs (see [21]) in their solution to the Heawood map-coloring problem. There are two possible cyclic orderings of each trivalent vertex. Under this convention, w... |